A blog for the mathematically curious

Möbius Strip: One-Sided Wonder

Whenever I introduce kids to möbius strips, I start by showing them the story of Wind and Mr. Ug by Vi Hart. I don’t tell them anything about it in advance, because I want to see if any of them can tell me what is happening. Sometimes, I give them a hint by writing Wind (written in the same style as she does in the video) on a thin piece of paper using a marker that bleeds through. I then flip the paper over.

Of course, Wind lives on a Möbius strip – a one-sided, one-edged surface, named after the German mathematician August Ferdinand Möbius (although it was independently discovered by another German mathematician, Johann Benedict Listing, the same year). This amazing surface can be made from a 2-sided strip of paper. Simply make a loop with the paper strip, then flip one side over before taping the edges together. There are several activities that can be used to explore the properties of the Möbius strip:

Draw a line, in the middle, down the length of the strip. Keep drawing and your line meets itself, showing that the strip has only one side.

Draw a line down the edge of the strip (like Wind building her fence). Again, the line will meet itself. The line will be along both sides of the strip, showing that the strip has only one edge.

Cut the strip down this center line, to see if Wind will see her dog again. (My son and I disagree about this. He argues that they will be on opposite sides of the new (non-Möbius) strip. I argue that since Wind’s world is transparent, it doesn’t matter.)

Some other fun things to try:

After cutting the strip down the center-line, cut down the center-line of the new strip.

Start with a new Möbius strip, and instead of cutting down the center-line, cut a third of the way from the edge.

If you don’t have any paper or scissors (or you’re just too lazy to get up from the computer – I’m not judging!), you can see the results of some of these Möbius strip experiments in this video: